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A020704 Pisot sequences E(3,10), P(3,10). 1
3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, 467280, 1543321, 5097243, 16835050, 55602393, 183642229, 606529080, 2003229469, 6616217487, 21851881930, 72171863277, 238367471761, 787274278560, 2600190307441, 8587845200883, 28363725910090 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT] (2016)

FORMULA

a(n) = 3*a(n-1) + a(n-2) (holds at least up to n = 1000 but is not known to hold in general).

Conjectures from Colin Barker, Jun 05 2016: (Start)

a(n) = (2^(-1-n)*((3-sqrt(13))^n*(-11+3*sqrt(13)) + (3+sqrt(13))^n*(11+3*sqrt(13))))/sqrt(13).

G.f.: (3+x) / (1-3*x-x^2).

(End)

Theorem: For E(3,10), a(n) = 3 a(n - 1) +  a(n - 2) for n>=2. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016

MATHEMATICA

RecurrenceTable[{a[0] == 3, a[1] == 10, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 30}] (* Bruno Berselli, Feb 05 2016 *)

PROG

(MAGMA) Exy:=[3, 10]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..30]]; // Bruno Berselli, Feb 05 2016

CROSSREFS

This is a subsequence of A006190.

See A008776 for definitions of Pisot sequences.

Sequence in context: A271943 A255116 A006190 * A289450 A113299 A126931

Adjacent sequences:  A020701 A020702 A020703 * A020705 A020706 A020707

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified August 21 21:36 EDT 2017. Contains 290908 sequences.